Universal properties of growing networks

Networks growing according to the rule that every new node has a probability pk of being attached to k preexisting nodes, have a universal phase diagram and exhibit power-law decays of the distribution of cluster sizes in the non-percolating phase. The percolation transition is continuous but of infinite order and the size of the giant component is infinitely differentiable at the transition (though of course non-analytic). At the transition the average cluster size (of the finite components) is discontinuous.